Thursday, July 31, 2014

I was clearing out some old worksheets that I either have stored somewhere electronically or that I know that I'll just never use again, given all the material that I have available to me, stored electronically, when I found the following that I picked up off a student's desk years ago. If I ever knew who the student was (and I probably did at the time), I've forgotten. He (possibly, but not likely, "she") didn't sign the work, nor put their name at the top of the sheet of loose leaf, as they have a tendency to do.

This is a bonus, not a Guest Artist, and isn't numbered along with my other (x, why?) strips.

I hope it's not a commentary on the situation in my classroom. More a commentary on social life: keep your head down and you won't get hurt. Or something.

Wednesday, July 30, 2014

Removed from circulation from my school library and regulated to the trash pile before I spotted the circa-1960 photo of Isaac Asimov on the back cover (looking younger than I've ever seen him before, so I surprised myself by recognizing him), serendipity brought Realm of Measure: From the yardstick to the Theory of Relativity into my possession.

I'm happy that it did. Though billed as an exploration of mathematics, he veers off a bit into scientific measurements, but I'll still count this toward my goal of reading one interesting book on math each summer, and this one does it without spinning out of control with endless, overly-complicated and overly-ridiculous equations.

Asimov goes into the history of measurements and how certain units came about and how the different units relate to one another. Not only did lengths like palms and feet have to be standardized from person to person and town to town, but also in relation to each so that they could be divided more evenly among people without formal education but who could count and compute the basic operations.

Asimov pushes for the metric system often throughout the book, as it's used a lot in science (where he was quite at home), not to mention in most of the non-English-speaking world. (He doesn't actually mention the measure of the English-speaking world using British units.)

The biggest problem with his arguments is that he presented the beauty of American/British system in its origins. If you were a wordsmith, you might be interested in the etymology of words, where they came from and how they came to be. You wouldn't stand for simplified spellings that are attempted from time to time. (Benjamin Franklin had a serious plan to change the language and simplify spelling, for instance.) When you read the origin of who's foot we use and who's armlength, and why a furlong is 1/8 of a mile, there is a wonder to it that goes beyond, "You see, there's this stick, and it has these two marks in it...."

Further, the system of divisions make sense. Think of the times. Think of the people and how they lived. If they split things, they likely halved them. If they had to quarter something, they halved it again. How often did someone come along with nine of his friends and need things sorted out evenly among them. And for all those divisions, 10 isn't a great number to work with: you can only divide it by 2 and 5, but not 3 nor 4. Dividing 12 by 2, 3, 4 and 6 proved more convenient, if it you lose 5. Moreover, metric conversion is easy in that you can switch units simply by moving the decimal point, but first you had to invent the decimal point! And that didn't happen to, what, the sixteenth century?

Oddly enough, I can sit here and argue that the time for the metric system has passed. We're living in a computer age, ruled by binary and hexidecimal. The number 10 really doesn't fit well into that scheme. And once you get passed 11th year math, base 10 goes out the window in favor of natural logs and e.

Not that any of this took away from my enjoyment of his book, which I heartily recommend to all with the proviso, "Don't try to read it in bed when you're really tired."

And I'll close by considering how close NYC came to allowing 16-ounce soft drinks while banning 500 milliliters. Metric: not even once.

Interesting (to me) Trivia!: This is comic #888 and it's appearing on 7/28/14, a day of sevens! Not that this was planned, my Internet was actually out for a day or so last week, and I wasn't feeling all that great anyway....

Saturday, July 19, 2014

The following are a bunch of links from pages I ripped out of teacher magazines. I haven't visited any of them despite having these pages for a very long time. So I'm sharing them here so I can look at them sometime in the future and clean up some of the random clutter on my desk at home.

CC.BetterLesson.com/mtp: More than 3000 classroom-ready lessons that can be integrated into any curriculum.
Lessons created by more than 130 Master Teachers, with step-by-step instructions, videos, thoughts on how the strategies help to implement Common Core, sample work, etc.

Share My Lesson: "Numerous resource aligned to the Common Core State Standards". A reminder that you are not alone.

10 Free Things! A page from NEA.org, where you can find books, activities, posters, lessons, videos, and more (or so they say!).
Some of the 10 things listed

Sunday, July 06, 2014

When is a Parallelogram not also a Rhombus? That's not a riddle, although if I think of a good punchline for it, I could make a comic out of it.
No, this is a serious Geometry question. In fact, it was the proof on the June 2014 Geometry Regents exam.

Specifically, question 38 read as follows:

The vertices of quadrilateral JKLM have coordinates J(-3,1), K(1,–5), L(7,-2), and M(3,4).
Prove that JKLM is a parallelogram.
Prove that JKLM is not a rhombus.

EDIT: Co-ordinates fixed.EDIT: Link added -- June 2014 Geometry Regents, question 38.Additionally, a grid was provided, but its use was optional. What wasn't given was enough space to write the answer, unless you thought to flip to the following page. The good news was that you did NOT have to write a two-column proof, but you did have to show your work and give the details.

This is a question my students should be able to answer easily. They probably didn't, but they should have. There are reasons why I say this.

When I approach any question that contains the word "Prove", I try to get the students to think of any TV show courtroom drama they've ever scene. (Sit-coms are a totally different animal, here.) Few of them might know what it's like to be in a real courtroom, so I settle for the simplified version. Each side will give an opening statement, and they will state what they are setting out to prove. They were present their evidence and build a case out of all the evidence they bring forward. There isn't one magic witness or exhibit that will hand-wave the case away. (If there were, the case likely would never have been brought.) Then, in the end, there are summaries, which include the lawyers' conclusions based on the evidence that they've presented, and they implore the jury to reach the same conclusion.

Students are different. Students, who a year earlier in Algebra would have me solve a five-step problem in 17 steps, and repeat the solution and check twice because they still weren't "getting it", will look at a Geometry "proof" and say, "It's true because, you know, Math."

They're not sure what the "Math" is, but the Math is there, so it must be true.
The point is that they need to be sure.

This led to some interesting discussions online about the test question. How do you know, for example, if you gave enough information for only 2 points and not for 4 points?

Let's get to the basics

Parallelograms and Rhombuses

Prove that JKLM is a parallelogram. Prove that JKLM is not a rhombus.

How do you prove that a quadrilateral is also a parallelogram?

You have several choices: if the opposite sides are parallel, or if the opposite sides are congruent, or if the diagonals bisect each other, then the figure is a parallelogram.
This statement, whichever one or ones you use, has to be your conclusion, but you have to back those up with evidence:

To prove opposite sides are parallel, you need to find the slopes of the four sides

To prove that the opposite sides are congruent, you need to find the lengths of the four sides

To prove that the diagonals bisect each other, you have to find the midpoints of both diagonals.

Any of those are easy to do, although I'd say that slopes and midpoints are quicker to find than the lengths. Which one you do is up to you, but you can save time if you keep the second part of the question in mind. Using the slopes for the parallelogram is fine, but it won't help you with the rhombus.

That being said, I'd probably find the slopes first just because it's pretty much second-nature to me to do that first.

How do you prove that a quadrilateral is rhombus?

You have a couple of choices: if all four sides are congruent, or if the diagonals perpendicularly bisect each other, then the figure is a rhombus.
This statement, whichever one or ones you use, has to be your conclusion, but you have to back those up with evidence:

To prove that the opposite sides are congruent, you need to find the lengths of, at least, two consecutive sides. If you know it's a parallelogram, then you know that the opposite sides are congruent. It would've helped if you did the distance formula before now, instead of slopes, but you aren't penalized for doing extra work.

To prove that the diagonals perpendicularly bisect each other, you have to find the slopes of the diagonals. If you know it's a parallelogram, then you know that the diagonals bisect each other. You don't need to prove this, even if you didn't show it earlier.

So how much work is actually required for this problem?

Believe it or not, you could have solved this problem simply by finding the lengths of the four sides. Is that worth 6 points? No, that was worth 2 points. The rest of the points came from you conclusions and your reasons. Just because you found the lengths, you haven't (and this isn't meant to be a plotting reference) you haven't connected the dots yet. You haven't given a conclusion, nor stated under what rules this proves your case.

JKLM is a parallelogram because the opposite sides are congruent. (Work is shown for this.) JKLM is NOT a rhombus because all four sides are NOT congruent.

or

JKLM is a parallelogram because the diagonals bisect each other. (Work is shown for this.) JKLM is NOT a rhombus because the diagonals are not perpendicular. (Work is shown for this.)

Either of these would be complete answers good for full credit.

By contrast "They're parallel." ... well, just isn't.

As you can see, the slopes of the sides aren't needed for the problem, but it isn't incorrect to find them. Other work left for the reader: find the lengths of the sides, and the midpoints and slopes of the diagonals.

About Me

Mr. Burke is a high school math teacher in New York as well as a part-time writer, and a fan of science-fiction/fantasy books and films.
He started making his own math webcomic totally by accident as a way of amusing his students and trying to make them think just a little bit more.
Unless otherwise stated, all math cartoons and other images on this webpage are the creation and property of Mr. Chris Burke and cannot be reused without permission.
Thank you.